Spectral properties of elliptic operator with double-contrast coefficients near a hyperplane

In this paper we study the asymptotic behaviour as $\epsilon \to 0$ of the spectrum of the elliptic operator $$\mathcal{A}$^\varepsilon=-{1\over b^\varepsilon}$\mathrm{d}\mathrm{i}\mathrm{v}$(a^\varepsilon\nabla)$posedinaboundeddomain$\Omega\subset$\mathbb{R}$^n$ $(n\ge 2)$ subject to Dirichlet boundary conditions on $\mathrm{\partial }\mathrm{\Omega }$. When $\epsilon \to 0$ both coefficients $a^{\epsilon }$ and $b^{\epsilon }$ become high contrast in a small neighborhood of a hyperplane $\mathrm{\Gamma }$ intersecting $\mathrm{\Omega }$. We prove that the spectrum of ${\mathcal{A}}^{\epsilon }$ converges to the spectrum of an operator acting in $L^2(\mathrm{\Omega })\oplus L^2(\mathrm{\Gamma })$ and generated by the operation $-\mathrm{\Delta }$ in $\mathrm{\Omega }\setminus \mathrm{\Gamma }$, the Dirichlet boundary conditions on $\mathrm{\partial }\mathrm{\Omega }$ and certain interface conditions on $\mathrm{\Gamma }$ containing the spectral parameter in a nonlinear manner. The eigenvalues of this operator may accumulate at a finite point. Then we study the same problem, when $\mathrm{\Omega }$ is an infinite straight strip (“waveguide”) and $\mathrm{\Gamma }$ is parallel to its boundary. We show that ${\mathcal{A}}^{\epsilon }$ has at least one gap in the spectrum when $\epsilon$ is small enough and describe the asymptotic behaviour of this gap as $\epsilon \to 0$. The proofs are based on methods of homogenization theory.